We approach the questions, what part of evolutionary change results from selection, and what is the adaptive information flow into a population undergoing selection, as a problem of quantifying the divergence of typical trajectories realized under selection from the expected dynamics of their counterparts under a null stochastic-process model representing the absence of selection. This approach starts with a formulation of adaptation in terms of information and from that identifies selection from the genetic parameters that generate information flow; it is the reverse of a historical approach that defines selection in terms of fitness, and then identifies adaptive characters as those amplified in relative frequency by fitness. Adaptive information is a relative entropy on distributions of histories computed directly from the generators of stochastic evolutionary population processes, which in large population limits can be approximated by its leading exponential dependence as a large-deviation function. We study a particular class of generators that represent the genetic dependence of explicit transitions around reproductive cycles in terms of stoichiometry, familiar from chemical reaction networks. Following Smith (2023), which showed that partitioning evolutionary events among genetically distinct realizations of lifecycles yields a more consistent causal analysis through the Price equation than the construction from units of selection and fitness, here we show that it likewise yields more complete evolutionary information measures. 

Beyond fitness: The information imparted in population states by selection throughout lifecycles.

We consider problems in the functional analysis and evolution of combinatorial chemical reaction networks as rule-based, or three-level systems. The first level consists of rules, realized here as graph-grammar representations of reaction mechanisms. The second level consists of stoichiometric networks of molecules and reactions, modeled as hypergraphs. At the third level is the stochastic population process on molecule counts, solved for dynamics of population trajectories or probability distributions. Earlier levels in the hierarchy generate later levels combinatorially, and as a result constraints imposed in earlier and smaller layers can propagate to impose order in the architecture or dynamics in later and larger layers. We develop general methods to study rule algebras, emphasizing system consequences of symmetry; decomposition methods of flows on hypergraphs including the stoichiometric counterpart to Kirchhoff’s current decomposition and work/dissipation relations studied in [1]; and the large-deviation theory for currents in a stoichiometric stochastic population process, deriving additive decompositions of the large-deviation function that relate a certain Kirchhoff flow decomposition to the extended Pythagorean theorem from information geometry. The latter result allows us to assign a natural probabilistic cost to topological changes in a reaction network of the kind produced by selection for catalyst-substrate specificity. We develop as an example a model of biological sugar-phosphate chemistry from a rule system published in [2]. It is one of the most potentially combinatorial reaction systems used by biochemistry, yet one in which two ancient, widespread and nearly unique pathways have evolved in the Calvin-Benson cycle and the Pentose Phosphate pathway, which are additionally nearly reverses of one another. We propose a probabilistic accounting in which physiological costs can be traded off against the fitness advantages that select them, and which suggests criteria under which these pathways may be optimal.

Rules, hypergraphs, and probabilities: the three-level analysis of chemical reaction systems and other stochastic stoichiometric population processes

We delve into growing open chemical reaction systems (CRSs) characterized by autocatalytic reactions within a variable volume, which changes in response to these reactions. Understanding the thermodynamics of such systems is crucial for comprehending biological cells and constructing protocells because it sheds light on the physical conditions necessary for their self-replication. Building on our recent work, where we developed a thermodynamic theory for growing CRSs featuring basic autocatalytic motifs with regular stoichiometric matrices, we now expand this theory to include scenarios where the stoichiometric matrix has a nontrivial left kernel space. This extension introduces conservation laws, which limit the variations in chemical species due to reactions, thereby confining the system's possible states to those compatible with its initial conditions. By considering both thermodynamic and stoichiometric constraints, we clarify the environmental and initial conditions that dictate the CRSs' fate—whether they grow, shrink, or reach equilibrium. We also find that the conserved quantities significantly influence the equilibrium state achieved by a growing CRS. These results are derived independently of specific thermodynamic potentials or reaction kinetics, therefore underscoring the fundamental impact of conservation laws on the growth of the system.
 
 
 
 
 Published by the American Physical Society
 2024
 
 


http://link.aps.org/pdf/10.1103/PhysRevResearch.6.023173

Thermodynamic and stoichiometric laws ruling the fates of growing systems

Autocatalysis is thought to have played an important role in the earliest stages of the origin of life. An autocatalytic cycle’s (AC) constituent chemicals can collectively catalyze their own recre-ation. When the reactions of multiple, interacting ACs are active in a region of space, they form an autocatalytic chemical ecosystem (ACE). Previous work demonstrated that, in chemostats, interactions between ACs in ACEs can be framed as analogous to those between species in biological ecosystems. Here, we extend this framework to investigate the effects of surface adsorption, desorption, and diffusion on ACE ecology. Simulating ACEs as particle-based stochastic reaction-diffusion systems in spatial settings, including open, two-dimensional reaction-diffusion systems and adsorptive mineral surfaces, we demonstrate that spatial structure can support more coexisting ACs and expose new AC traits to selection.

ACEs in spaces: Autocatalytic Chemical Ecosystems in Spatial Settings

The predator–prey model is a fundamental mathematical tool in ecology used to understand the dynamic relationship between predator and prey populations. This study develops a fractional-order delayed dynamical model for a four-species food web, which includes an intermediate predator feeding on two prey species and a top predator preying on all three species. The boundedness of the system’s solutions is first rigorously established using the Laplace transform method. Next, a nonlinear dynamical analysis is performed to determine the existence conditions and local stability of both the trivial and positive equilibrium points. In particular, by treating the time delay as a bifurcation control parameter, explicit criteria for the onset of Hopf bifurcation are derived. Theoretically, when the delay magnitude exceeds a critical threshold, the system loses stability and exhibits sustained oscillatory behavior. Finally, systematic numerical simulations are performed under specific parameter settings. The effects of varying fractional orders and delay magnitudes on the system’s dynamics are quantitatively explored, and the results show strong agreement with the theoretical predictions. 

Dynamics in a Fractional-Order Four-Species Food Web System with Top Predator and Delays

Polyhedral geometry and combinatorics of an autocatalytic ecosystem

Developing a mathematical understanding of autocatalysis in chemical reaction networks has both theoretical and practical implications. For a class of autocatalysis, which we term 'stoichiometric autocatalysis', we show that it is possible to classify them in equivalence classes and develop mathematical results about their behavior. We also provide a linear-programming algorithm to exhaustively enumerate them and a scheme to visualize their polyhedral geometry and combinatorics. We then define cluster chemical reaction networks, a framework for coarse-graining realistic chemical reactions using conservation laws. We find that the list of minimal autocatalytic subnetworks in a maximally connected cluster chemical reaction network with one conservation law grows exponentially in the number of species. We end our discussion with open questions concerning autocatalysis and multidisciplinary opportunities for future investigation.

pdf/the-geometry-and-combinatorics-of-an-autocatalytic-ecology-167hgtbi.pdf

The geometry and combinatorics of an autocatalytic ecology in chemical and cluster chemical reaction networks

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Polyhedral geometry and combinatorics of an autocatalytic ecosystem

The geometry and combinatorics of an autocatalytic ecology in chemical and cluster chemical reaction networks